Subgroups — #32150946

Order 18176.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597274	C₂	1	1	yes	maximal, characteristic
2	#22590248	C₂	1	4	yes	maximal, complemented
3	C₁₆×Dic₁₄₂	C₂	1	4	yes	maximal, complemented
4	#22590098	C₂	1	2	yes	maximal
5	#22590096	C₂	1	1	yes	maximal, characteristic
6	#22589936	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#17986556	C₄	1	1	yes	characteristic
9	C₂×#14556068	C₂²	1	2	yes	—
10	#17986129	C₄	1	4	yes	—
11	C₈×Dic₁₄₂	C₂²	1	4	yes	—
12	#17986025	C₄	1	2	yes	—
13	#17986023	C₄	1	1	yes	characteristic
14	#17985954	C₂²	1	2	yes	—
15	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	4	yes	—
16	Dic₇₁⋊C₁₆	C₂²	1	8	yes	—
17	#17985774	C₂²	1	4	yes	—
18	C₁₆×Dic₇₁	C₂²	1	4	yes	—
19	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
23	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
24	C₂×#11765849	C₂×C₄	1	2	yes	—
25	#14556128	C₈	1	2	yes	—
26	#14556120	C₈	1	2	yes	—
27	#14556107	C₈	1	4	yes	—
28	C₄×Dic₁₄₂	C₂×C₄	1	4	yes	—
29	#14556087	C₂×C₄	1	2	yes	—
30	#14556068	—	2	4	no	—
31	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
32	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
33	Dic₇₁⋊C₈	C₂×C₄	1	8	yes	—
34	#14556038	C₂×C₄	1	4	yes	—
35	C₈×Dic₇₁	C₂³	1	4	yes	—
36	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₇₁	—	2	2	no	—
40	C₂×C₁₆×C₇₁	Q₈	1	4	yes	complemented
41	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
42	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₁	C₂³	1	2	yes	—
46	C₂×Dic₁₄₂	C₂×C₈	1	4	yes	—
47	#11765849	—	2	4	no	—
48	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
49	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
50	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
51	Dic₇₁⋊C₄	C₂×C₈	1	8	yes	—
52	C₂²⋊Dic₇₁	C₂×C₈	1	4	yes	—
53	C₄×Dic₇₁	C₂²×C₄	1	4	yes	—
54	C₇₁⋊C₁₆	—	2	4	no	—
55	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₇₁	#41	1	2	yes	—
62	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
63	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
66	C₁₆×C₇₁	—	2	2	no	—
67	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
68	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
69	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₇₁	#131	1	2	yes	—
76	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₇₁	C₂²×C₈	1	8	yes	—
78	Dic₁₄₂	—	2	8	no	—
79	C₈×C₇₁	—	2	2	no	—
80	C₈×C₇₁	C₄×Q₈	1	2	yes	—
81	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
83	C₇₁⋊C₈	—	2	4	no	—
84	C₂²×C₇₁	—	2	2	no	—
85	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
86	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₇₁	—	2	2	no	—
88	C₄×C₇₁	C₈×Q₈	1	2	yes	—
89	C₄×C₇₁	#504	1	4	yes	—
90	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
92	Dic₇₁	—	2	8	no	—
93	#33474	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	—	2	2	no	—
95	C₂×C₇₁	C₂×C₈×Q₈	1	1	yes	characteristic
96	C₂×C₇₁	C₂×#504	1	2	yes	—
97	#2960	—	71	1	no	—
98	#2183	—	71	4	no	—
99	Q₈×C₁₆	—	71	4	no	—
100	#2151	—	71	2	no	—
101	#2150	—	71	1	no	—
102	#2108	—	71	2	no	—
103	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
104	C₇₁	#33474	1	1	yes	characteristic, complemented, Sylow
105	#533	—	71	1	no	—
106	C₂×M₃₂	—	71	2	no	—
107	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
108	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
109	#446	—	71	4	no	—
110	C₈×Q₈	—	71	4	no	—
111	#424	—	71	2	no	—
112	#423	—	71	1	no	—
113	#405	—	71	2	no	—
114	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
115	C₄⋊C₁₆	—	71	8	no	—
116	C₄⋊C₁₆	—	71	4	no	—
117	#346	—	71	4	no	—
118	C₄×C₁₆	—	71	4	no	—
119	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
120	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
121	C₂×M₁₆	—	71	2	no	—
122	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
124	C₄⋊Q₈	—	71	2	no	—
125	#125	—	71	2	no	—
126	C₂²⋊Q₈	—	71	4	no	—
127	C₄×Q₈	—	71	4	no	—
128	#117	—	71	2	no	—
129	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
130	M₃₂	—	142	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	71	4	no	—
133	C₂×C₁₆	Dic₁₄₂	1	4	yes	complemented
134	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
135	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
136	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
137	C₄⋊C₈	—	71	8	no	—
138	C₄⋊C₈	—	71	4	no	—
139	#97	—	71	4	no	—
140	C₄×C₈	—	71	4	no	—
141	C₄×C₈	C₄×D₇₁	1	2	yes	—
142	C₄×C₈	C₂²×D₇₁	1	2	yes	—
143	C₂×Q₈	—	71	4	no	—
144	C₂²×C₄	C₈×D₇₁	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
146	M₁₆	—	142	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	71	4	no	—
149	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
151	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
152	C₂×C₈	#11765865	1	2	yes	—
153	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
154	C₄⋊C₄	—	71	8	no	—
155	C₄⋊C₄	—	71	4	no	—
156	C₄⋊C₄	—	71	4	no	—
157	C₂²⋊C₄	—	71	4	no	—
158	C₄²	—	71	4	no	—
159	C₄²	C₈×D₇₁	1	1	yes	characteristic
160	C₄²	C₈×D₇₁	1	1	yes	characteristic
161	C₄²	C₂×C₄×D₇₁	1	2	yes	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	142	4	no	—
164	C₁₆	C₂×Dic₁₄₂	1	2	yes	—
165	C₁₆	C₂×Dic₁₄₂	1	2	yes	—
166	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
167	Q₈	—	142	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	71	8	no	—
170	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
174	C₂×C₄	#14556134	1	2	yes	—
175	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	142	4	no	—
178	C₈	C₄×Dic₁₄₂	1	2	yes	—
179	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
180	C₈	C₂×#11765865	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₈×Dic₁₄₂	1	2	yes	—
183	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	142	8	no	—
186	C₄	C₈×Dic₁₄₂	1	2	yes	—
187	C₄	#17986415	1	4	yes	—
188	C₄	C₂×C₄×Dic₁₄₂	1	1	yes	characteristic
189	C₄	C₂×#14556134	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×C₈×Dic₁₄₂	1	1	yes	characteristic
192	C₂	C₂×#17986415	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#33474
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22590248	C₂	semidirect
3. C₁₆×Dic₁₄₂	C₂	semidirect
40. C₂×C₁₆×C₇₁	Q₈	semidirect
104. C₇₁	#33474	semidirect
133. C₂×C₁₆	Dic₁₄₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22597274	2
#22590248	2
C₁₆×Dic₁₄₂	2
#22590098	2
#22590096	2
#22589936	2
C₂×C₄×C₁₆×C₇₁	2
#33474	71